The selection of suitable EOSs results from a delicate balance between complexity and accuracy. To achieve this, a method of continuous transformation of EOS is introduced. It is governed by a contact Hamilton function f depending on all the extensive and intensive thermodynamic variables U, S, V, N, T, p and p. Associated to f are Mrugala's evolution equations which maintain the thermodynamic consistency [Rep. Math, Phys. 29 (1991) 109]. Hamilton function f may also depend on the evolution parameter, t [Methodes geometriques pour l'etude des systemes thermo-dynamiques et la generation d'equations d'etat, Ph.D. thesis, 1999]. By solving this differential system, starting from some state of a system at t = 0, we obtain a state of a new system, at t = 1. Thus, the EOSs of the new system can be deduced from those of the initial one and from the f chosen. The main theoretical result is that contact Hamilton functions can be deduced from the thermodynamic potentials. To illustrate this, several f are built, which map an ideal gas to a van der Waals fluid, yield Peneloux volume shift method or produce Lee-Kesler (LK) method of interpolation. This approach provides a systematic method for organizing and relating EOSs.